The paper compares two battles in the same area, one (480 BCE) in which historical accounts say a Greek colony was defended by a coalition of Greek allies, and another, later battle (408 BCE), in which it stood alone and was defeated.

They establish a baseline of what they expect local strontium isotope composition should be both from local flora and fauna samples and from burials not associated with the battle. They then compare this to the greek combatants of these two battles. They find that 67% of Greek soldiers from 480 BCE are likely non-local, while 36% of Greek soldiers from 408 BC were non-local.

Up to this point, the paper supports historical sources and seems to be quite well done. After this point, they speculate that the non-local individuals present were mercenaries based solely on the strontium isotope ratios in their tooth enamel, and further speculate on where specific individuals might have originated. This is where the authors might be going out on a limb.

Greek colonies were widespread at this point in history, and it's very likely that some "mercenaries" were just from distant colonies. This doesn't seem to be adequately addressed in the paper. It's also possible that soldiers ate a different diet than the locals, possibly higher in fish or imported foods, which could have made them appear non-local. It's also possible soldiers moved around more, and that might have affected their levels in difficult to predict ways.

Bottom line, strontium isotope analysis is great at the population level, but there's a lot that can go wrong when dealing with individuals. I find the notion that Greek's hired mercenaries plausible, but this paper doesn't provide strong evidence for it.